The question asks you to graph the function $h(x) = (x + 2)^2 + 5$ by using transformations of the standard quadratic function $f(x) = x^2$.

Step-by-Step Solution:

1. Start with the base function: $f(x) = x^2$, which is a parabola opening upwards with its vertex at the origin (0, 0).

2. Apply the horizontal shift: The expression $(x + 2)$ indicates a horizontal shift. Since it is $x + 2$, the graph shifts 2 units to the left.

3. Apply the vertical shift: The $+5$ outside of the squared term indicates a vertical shift upwards by 5 units.

Determining the Correct Graph:

• The vertex of the new function $h(x) = (x + 2)^2 + 5$ will be at $(-2, 5)$.
• The graph should be a parabola that opens upwards, shifted left 2 units and up 5 units from the original $x^2$ graph.

Looking at the Options:

• Option A: Shows a graph with the vertex at (-2, 5), which matches the transformation.
• Options B, C, and D: Do not match the correct shifts of left 2 units and up 5 units.

Conclusion:

The correct answer is A.

Would you like a deeper explanation of transformations or have any other questions? Here are some related questions that could help you understand transformations better:

1. What are the different types of transformations for graphs?
2. How do you determine the direction of shifts from a function's equation?
3. What is the effect of multiplying a function by a negative number?
4. How does a vertical stretch or compression affect the graph of a function?
5. What is the difference between a horizontal shift and a vertical shift?

Tip: Remember that a positive number inside the parentheses shifts the graph to the left, and a positive number outside shifts it upwards.